Invariance Times and BSDEs Stopped Before aRandom TimeStéphane Crépey (mainly based on joint work with Shiqi Song)
LaMME, Univ Evry, CNRS, Université Paris-Saclay
With the support of the “Chair Markets in Transition” (Fédération BancaireFrançaise)

Cours Bachelier, IHP, janvier–février 2018

1 / 36Based onCrépey, S. and S. Song (2017).
Invariance times.
Annals of Probability.
Crépey, S. and S. Song (2015).
BSDEs of counterparty risk.
Stochastic Processes and their Applications.
Albanese, C. and S. Crépey (2017).
XVA analysis from the balance sheet
Working paper on my webpage
Crépey, S. and S. Song (2016).
Counterparty risk and funding: Immersion and beyond.
Finance and Stochastics (long preprint version on my webpage).
Works in progress..
2 / 36

Notation

For any process Y and times T and θ
Y T = Y 1[0,T ) + YT 1[T ,+∞)
Assuming Y left-limited:Y θ− = Y 1[0,θ) + Yθ− 1[θ,+∞)Y θ−∧T = (Y θ− )T = (Y T )θ− ,SI (G, Q), MI (G, Q) Semimartingales, local martingales on a predictableinterval I
I = R+ by default

3 / 36

Mathematical viewpoint

Finding conditions under which the progressive enlargement of filtrations
Jeulin-Yor formula can be compensated by the Girsanov formula of an equivalentchange of probability measure

4 / 36

Counterparty risk motivation

Counterparty risk is related to cash flows or valuations linked to either
counterparty default or the default of the bank itself.
C. Albanese and S. C. XVA analysis from the balance sheet: A key distinction
is between
the cash flows received by the bank prior its default time θ
the cash flows received by the bank during the default resolution period
starting at θ.

The first stream of cash flows affects the bank shareholders, whereas thesecond stream of cash flows only affects creditors.

5 / 36

For accepting a new deal, shareholders need to be at least indifferent given
the cash flows before θ only.
→ In the context of so-called XVA analysis, all the cumulative cash flow and gain
processes need to be stopped before the default time θ of the bank itself.
→ XVA BSDEs stopped before a terminal time

6 / 36

Full BSDEGiven
a fixed time horizon (final maturity) T > 0
a totally inaccessible stopping time θ,a running cost gt (ω, y ),we consider the following BSDE in Y ∈ S(G, Q): YT 1{T <θ} = 0,(1)

R t∧θ∧T
Ytθ−∧T + 0
gs (Ys− )ds ∈ M(G, Q).

7 / 36Extensions to BSDEs
with g = gt (y , z), where additional arguments z correspond to integrands ina stochastic integral representation of the martingale part of Y
with further finite variation (non-necessarily Lebesgue absolutely continuous)
driving terms...
“Duffie, Schroder, and Skiadas (1996)’s” solution
Forget θ in (1) (“send it to infinity”), obtain a solution Ȳ of the resulting(simpler) equation and then set Y = Ȳ θ−
Only yields a solution Y to (1) if Ȳ does not jump at θ

8 / 36

Reduction of filtration

Let θ be a G stopping time (not necessarily totally inaccessible), and let F be a
subfiltration of G (F and G both satisfying the usual conditions), such that:

Condition (B)For any G predictable process L, there exists an F predictable process L0 , which
we call the F predictable reduction of L, such that L0 coincides with L until θ.

9 / 36

Lemma 1 (“If” part = Lemma 1 in Jeulin and Yor (1978))The filtration F satisfies the condition (B) if and only if G is a subfiltration of
F = (F t )t∈R+ , whereF t = {B ∈ A : ∃A ∈ Ft , A ∩ {t < θ} = B ∩ {t < θ}}.

10 / 36

The condition (B) relaxes the standard progressive enlargement of filtration
setup where G reduces to the smallest filtration containing F for which θ is a
stopping time
All the classical results of progressive enlargement of filtration are still valid
under (B):“Key lemma of credit risk”,
Existence of F optional reductions (also denoted with 0 ) coinciding with G
optional processes before θ,...

11 / 36

In particular, let S represent the F Azéma supermartingale of θ, i.e.
St = Q(θ > t |Ft ), t > 0, with Doob-Meyer decomposition S = Q − D.
The Jeulin-Yor theorem says that for any bounded (F, Q) martingale X , theprocessX θ− −

1(0,θ](F,Q)
 hS, X iS−

is a (G, Q) uniformly integrable martingale.

12 / 36

The direct part in the next result addresses the “inverse problem” of knowing
when an (F, Q) semimartingale X is such that X θ− is a (G, Q) local martingale.

Lemma 2 (Song (2014))
For any X ∈ S{S− >0} (F, Q) such that S−  X + [S, X ] ∈ M{S− >0} (F, Q),
then X θ− ∈ M(G, Q).
Conversely, for any M ∈ M(G, Q) with ∆θ M = 0, then M 0 ∈ S{S− >0} (F, Q)
and S−  M 0 + [S, M 0 ] ∈ M{S− >0} (F, Q).
“Immersion case” (of a pseudo-stopping time avoiding F stopping times)
where S is continuous and nonincreasing: Then [S, ·] = 0, so that themartingale conditions in the above reduce to X , resp. M’ ∈ M{S− >0} (F, Q).

13 / 36

The Jeulin formula and Lemma 2 can be viewed as progressive enlargement
formal analogs of the predictable and optional Girsanov measure change
formulas, the Azéma supermartingale S playing the role of the measurechange density from the probability measure Q to some P  Q
Predictable Girsanov formula“For any bounded X ∈ M(F, Q), X − S1−  hS, X i(F,Q) ∈ M(F, P)”
Optional Girsanov formula“X ∈ M(F, P) iff X ∈ S{S− >0} (F, Q) and S−  X + [S, X ] ∈ M{S− >0} (F, Q)”

14 / 36

Recall S = Q − D denotes the (F, Q) canonical Doob-Meyer decomposition of
St = Q(θ > t |Ft ), t > 0.

Lemma 3 (Song (2014))
One has the following unique predictable multiplicative decomposition of theAzéma supermartingale S of θ on {pS > 0}:
S = S0 E(−

11 D)E( p  Q).
S
S−

15 / 36Reduced BSDE

Recall the full BSDE with running cost g and terminal time θ (if < T )Letting U = Y 0 , the ”(G, Q) local martingale-to be” in the full BSDE satisfies
Ytθ−∧T +Z t∧θ∧T

gs (Ys− )ds = Utθ−∧T +

0

=



UT +
|

Z t∧θ∧T

gs0 (Us− )ds

0

Z ·∧T

θ−

0

t

T
gs0 (Us−
)ds
{z
}

.

U

This suggests to solve the full BSDE with Lemma 2.

16 / 36Namely, we consider the following BSDE for U ∈ S{S− >0} (F, Q):
UT ST = 0 , S−  U + [S, U] ∈ M{S− >0} (F, Q).

(2)

Proposition 1
The BSDEs (1) and (2) are equivalent. Specifically:• If Y is a solution to the BSDE (1), then U = Y 0 is a solution to the BSDE (2).
• Conversely, if U is a solution to the BSDE (2), then Y = U θ− is a solution to
the BSDE (1).
“Immersion case”, where S is continuous and nonincreasing, of a pseudo-stopping
time avoiding F stopping times
Then [S, ·] = 0, so that the martingale condition in (2) reduces to
U ∈ M{S− >0} (F, Q).

17 / 36

Condition (A)However, beyond this “immersion” case, even if passing from (1) to (2)allows removing θ from the equation, this comes at the expense of a more
involved martingale condition in (2)As a shortcut out of this, suppose

Condition (A)There exists a probability measure P equivalent to Q on FT such that, for any(F, P) local martingale P, P θ− is a (G, Q) local martingale on [0, T ].
Then any solution U ∈ S{S− >0} (F, P) to the “reduced BSDE”
UT ST = 0 , U t = UtT +Z t∧T

gs0 (Us− )ds ∈ M{S− >0} (F, P)(3)0yields a solution Y = U θ− to (1).
Noting Sθ− > 0 on {0 < θ < ∞}18 / 36

Invariance times

This allows getting rid of θ in the BSDE
Modulo reduction, the martingale condition in the reduced BSDE (3) isexactly the same as the one in the full BSDE (1).
As the running costs are integrated until the end of times T in the reduced
BSDE, this results in pricing rules consistent with the prescription of theregulator, which says quite explicitly that a bank capital cannot be seen
increasing as a consequence of the sole deterioration of the bank credit, allelse being equal
UCVA versus first-to-default CVA

Definition 1
If the condition (A) is satisfied, we call the random time θ an invariance time andthe related probability measure P an invariance probability measure.

19 / 36

Stopping before θ rather than at θ in the condition (A) appears naturally in
our counterparty risk application and BSDE motivation.
However, the condition (A), with the “stopping before θ” operator, isnonstandard in the enlargement of filtration literature.
→

Strength of the condition (A)?

→

Equivalence, under the condition (A), between the full BSDE (1)and the reduced BSDE (3)??

20 / 36

Main result
Given a probability measure P equivalent to Q on FT , let q = p1 denote the (F, Q)
dPmartingale density function dQ

Ft∧T, t ∈ R+ .

Theorem 1
Assuming the condition (B) on F and given a constant T > 0:(i) A probability measure P equivalent to Q on FT is an invariance probability
measure if and only if
q = q0 E( p1S  Q) on {pS > 0} ∩ [0, T ](4)(ii) The condition (A) holds if and only ifE(1{pS>0} p1S  Q) is a positive (F, Q) true martingale on [0, T ].

(5)

In this case, an invariance probability measure P is given by the Q density1E(1{pS>0} p Q)TS

(6)21 / 36Proof idea. The measure change “compensates” the reduction of filtration.
Start with, for any P ∈ M(F, P)
1(0,θ]
 hQ, P − q  [p, P]i ∈ M[0,T ] (G, Q).
(P − q  [p, P])θ− −
{z
}
|
S−
Girsanov|{z}
Jeulin−Yor

But
1(0,θ]
 hQ, P − q  [p, P]iS−
1(0,θ]= P θ− − 1[0,θ) q  [p, P] + hQ, P − q  [p, P]i .
S−(P − q  [p, P])θ− −Hence P is an invariance probability measure iff
1[0,θ) q  [p, P] +
1(0,θ]
 hQ, P − q  [p, P]i ∈ M[0,T ] (G, Q) , ∀P ∈ M(F, P)
S−

Then project, use density arguments, etc.
22 / 36Proposition 2
If F = G and θ has an intensity, then θ cannot be an invariance time unlessQ(θ ≤ T ) = 0.
Given F ⊆ G satisfying (B), P = Q is an invariance probability measure for all
T > 0 if and only if Q = S0 .
Proof. 1) In the case where F = G and θ has an intensity, we have
S = 1[0,θ) , D is continuous , p S = 1[0,θ] , Q = 1[0,θ) + D and Q0 = S0 = 1.
Hence, using the stochastic exponential formulaY
1(1 + ∆s Q)e −∆s Q
E(1{pS>0} p  Q)t =E(Q)t = e Qt −Q0
S
s≤t

=e

1{t<θ} +Dt −1

Qt1{t<θ} = e 1{t<θ} ,which vanishes at θ on {θ ≤ T }. Therefore, in view of Theorem 1, the condition(A) cannot hold on [0, T ] unless Q(θ ≤ T ) = 0.
2) In the case where P = Q, we have q = q0 on [0, T ]. Hence, in view of Theorem
1, P is an invariance probability measure for all T > 0 if and only if Q is constanton [0, T ].
23 / 36

Example 1
Let G be the augmentation of the natural filtration of the jump process at an
exponential time θ relative to some probability measure Q.
For F = GThe condition (B) holds trivially
Proposition 2 1) shows that the condition (A) does not hold

For F trivial
As any G predictable process coincides with a Borel function before θ, thecondition (B) is satisfied.
The only (F = {∅, Ω}, Q) local martingales are the constants, so that P = Q is
an invariance probability measure and θ is an invariance time.

24 / 36

Theorem 1 shows that the condition (A) reduces to a mild integrability
condition.
In addition, Theorem 1 can be used to establish that, under the condition (A)(and assuming that θ has an intensity),
the full and reduced BSDEs are equivalent .

25 / 36Intensity based credit risk pricing formulas
Invariance times also allow extending the progressive enlargement credit risk
pricing formulas beyond the restrictive “immersion” (or pseudo-stopping times)
setup:

Theorem 2
Under the condition (A), if θ has an intensity γ and ST > 0:
For any nonnegative Ft measurable random variable χ:RT0E[χ1{T <θ} ] = EP [χe − 0 γu du ].

(7)

For any nonnegative F predictable process K ,Z T
Rs 0
Ks e − 0 γu du γs0 ds].
E[Kθ 1{θ≤T } ] = E [P

0→ Intensity models of credit risk with strong (adverse) dependence between
credit risk and the underlying market exposure
Wrong way and gap risk modeling
26 / 36

Connection with the survival measure

Rθ

Assuming that θ has a (G, Q) intensity γ such that e 0 γt dt is Q integrable,
Collin-Dufresne, Goldstein, and Hugonnier (2004)
introduce the “survivalR·
measure” S with the (G, Q) density process e 0 γt dt 1[0,θ) .
Under this measure S, Collin-Dufresne et al. (2004) are also able to derive arather general intensity based credit risk pricing formula, exempt from theDuffie et al. (1996)’s no-jump condition.

27 / 36

Theorem 3
Under
the condition (B), if ST > 0 and θ has a (G, Q) intensity γ such thatRθe 0 γt dt is Q integrable, then θ is an invariance time and the restriction to FT ofthe invariance probability measures P and of the survival measure S coincide.
Collin-Dufresne et al. (2004)’s solution touches the filtration as little aspossible but singularly changes the measure
Invariance times do the opposite and yield a full semimartingale calculus, notonly transfer of expectation formulas

28 / 36

Connection with pseudo-stopping times

This part gives examples which illustrate how the condition (A) can besatisfied in cases where (F, P) martingales really jump at θ, as well as theconnection between the condition (A) and the notion of pseudo-stopping
time in Nikeghbali and Yor (2005).
Consider a (0, +∞) valued random time θ. It is an (F, Q) pseudo-stopping
time if and only if X θ is a (G, Q) uniformly integrable martingale for any
bounded F martingale X (cf. Nikeghbali and Yor (2005)).
Clearly, if a pseudo-stopping time θ avoids the F stopping times, then it is aninvariance time satisfying the condition (A) for any positive constant T , withinvariance probability measure P = Q.

29 / 36

Let A denote the F dual optional projection of 1[θ,∞) . Nikeghbali and Yor(2005) show that θ is a pseudo-stopping time if and only if S = 1 − A.
Instead, Proposition 2 2) shows that P = Q is an invariance probability
measure for any positive constant T if and only if S = 1 − D
Noting that S0 = 1 here, as θ > 0.

Both conditions coincide if and only if A = D.
In the case where θ is a G totally inaccessible stopping time, A = D if andonly if θ avoids the F stopping times.
Hence, for a (0, +∞) valued G totally inaccessible stopping time θ, there are
two “orthogonal” cases:
If θ has the avoidance property, then θ is a pseudo-stopping time if and only if
Q is an invariance probability measure;If θ does not have the avoidance property, then θ cannot be a pseudo-stopping
time and Q be an invariance probability measure simultaneously.

30 / 36

The difference is due to the fact that a pseudo-stopping time is defined interms of stopping at θ, whereas invariance is defined in terms of stopping
before θ.
Having said this regarding the case where P = Q, we emphasize that, withrespect to a pseudo-stopping time that is defined with respect to the fixedprobability measure Q, the additional flexibility of invariance times lies in thepossibility to consider the martingale property under a changed measure P.
In fact, the pseudo-stopping time condition is very restrictive. By contrast
Theorem 3 shows that invariance times are the rule rather the exception.

31 / 36

Example 2 (An invariance time intersecting F stopping times..)
For i = 1, 2, let µi > 0 be a finite F stopping time with bounded
compensator vi . Assuming µ2 > T , define θ = 1A µ1 + 1Ac µ2 , whichintersects the F stopping times µi , for some A ∈ G∞ independent from F∞
such that α = Q(A) ∈ (0, 1).
On [0, T ], S = 1[0,µ1 ) α + 1[0,µ2 ) (1 − α) , S− ≥ 1 − α, andR ·∧θ 11
v = 0 Ss−
D is bounded. Therefore the conditions of TheoremdDs ≤ 1−α3 are fulfilled and θ is an invariance time.
Easy computations yield
A = (1[θ,∞) )o = 1[µ1 ,∞) α + 1[µ2 ,∞) (1 − α) , A∞ ≡ 1,
so that, by application of Theorem 1 (3) in Nikeghbali and Yor (2005), θ isalso a pseudo-stopping time.

32 / 36

Example 3 (..which is not a pseudo stopping time)Now, to obtain an invariance time θ intersecting F stopping times without
being a pseudo-stopping time, one can set
θ = 1A1 µ1 + 1A2 µ2 + 1A3 τ,
for a non pseudo-stopping time τ and a partition Ai , i = 1, 2, 3, independent
from F∞ and τ .
With αi = Q(Ai ) > 0, we have
A = (1[θ,∞) )o = α1 1[µ1 ,∞) + α2 1[µ2 ,∞) + α3 (1[τ,∞) )o ,where (1[τ,∞) )o∞ 6= 1, hence A∞ 6= 1, with positive Q probability. so that, by
the converse part in the above mentioned theorem, θ is not a
pseudo-stopping time.
But the Azéma supermartingale of θ is given by
S = 1[0,µ1 ] α1 + 1[0,µ2 ] α2 +o (1[0,τ ) )α3 ≥ α2 on [0, T ].
Hence, the other computations above do not change, which shows that θ isan invariance time.
33 / 36

Counterparty risk on credit derivatives

Copula model of θ0 , θ1 , . . . , θn , where θ0 = θ corresponds to the default timeof the counterparty of a bank in credit derivatives on names 1, . . . , nCounterparty risk computations: need make the model dynamic byintroduction of a suitable model filtration G
Can one separate the information that comes from θ0 from a reference
filtration F?
→ Reduction of filtration in this sense

34 / 36

For applications, some kind of martingale invariance property is required, but
under minimal assumptions, so that the model stays as flexible as possible inview of applications→ Invariance times→ Intensity models of counterparty risk with strong (adverse) dependence
between the credit risk of a counterparty and the underlying market (credit inthis case) exposure→ Wrong way and gap risk modeling

35 / 36

Dynamic copula models
Dynamic Marshall-Olkin copula (common-shock) model“Gap risk”
θ = θ0 is an invariance time(A) achieved with P = Q but G greater than the classical progressive
enlargement of F by θ

Dynamic Gaussian copula model with correlation parameter % ∈ [0, 1]“Wrong-way risk”
Gives together an example of a model where the invariance property issatisfied but immersion does not hold (i.e. P 6= Q), for small %And, for larger %, an example of a model where the invariance property may
not be satisfied

36 / 36Intensity Based Pricing Formulas, Survival Measure andInvariance Times: Discussion in a univariate DGC SetupLetθ = h−1Z +∞ς(u)dBu ,

(8)

0

where
ς is a Borel function on R+ such that
B is a standard Brownian motion,h = Φ−1 ◦ Eλ ,R +∞ 2ς (u)du = 1,
0or any continuously differentiable increasing function from (0, +∞) to R.

37 / 36

The reference filtration F is taken as the augmented filtration of the natural
filtration B of B.
The full model filtration G is given as the augmented filtration of the
progressive enlargement of F = B by θ.
Note that the G stopping time θ is F∞ measurable.

38 / 36

Let ht = 1{θ≤t} and
Z t

2

Z +∞

ς(u)dBu , kt = (ht , θ ∧ t) , ν (t) =mt =0ς 2 (v )dv ,(9)twith ν assumed positive for all t.

Lemma 4The process (m, k) is (G, Q) Markov.

39 / 36

Lemma 5
We haveQ(θ > t | Ft ) = Φh(t) − mt 
, t ∈ R+ ,
ν(t)

(10)

where Φ denotes the standard normal cdf.
Infinite variation → the reference filtration F = B is not immersed into thefull model filtration G.

40 / 36

Lemma 6
There exists processes of the form
µt = µ(t, mt , kt ) and γt = γ (t, mt , kt ) = γt 1(0,θ] , t ∈ R+ ,(11)

for continuous functions µ and γ with linear growth in m, such that
dWt = dBt − µt dt is a (G, Q) Brownian motion and
γ is the (G, Q) intensity of θ.

(12)41 / 36Proposition 3
Let a process m? satisfy
dmt? = ς(t) dWt? + β(t, mt? , (0, t))dt , 0 ≤ t ≤ T ,(13)starting from m0? = 0, for some Brownian motion W ? with respect to somestochastic basis (G? , Q? ). Denoting the Q? expectation by E? , we have, for any
positive constant T and bounded Borel function G (t, m),h
E 1θ<T G (θ, mθ ) = E?

Z T

i
Rt
?
e − 0 γ(s,ms ,(0,s)))ds γ(t, mt? , (0, t))G (t, mt? )dt .

(14)

0

Proof. by Feynman–Kac representations of PDE solutions.

42 / 36

A contrario, we expect that
hZ T
i
Rt


E 1θ<T G (θ, mθ ) 6= E
e − 0 γ(s,ms ,(0,s))ds γ(t, mt , (0, t))G (t, mt )dt

(15)

0

(except in special cases such as G = 0), because, from (9) and (11)–(12), it holdsdmt = ς(t) dWt + β(t, mt , kt )dt , t ∈ R+ ,(16)
which, for t ≥ θ so that kt = (1, θ), diverges from the specification (13).

43 / 36

In fact, let
hZ T
i
Rs
Vt = Ee − t γ(u,mu ,(0,u))du γ(s, ms , (0, s))G (s, ms )ds Gt , t ∈ R+ ,t

so that V0 is equal to the right hand side in (15).
By an application of Duffie et al. (1996, Proposition 1) with X = r = 0 and
h· = γ(·, m· , (0, ·)) on [0, T ] there, we have
E 1θ<T G (θ, mθ ) = V0 − E∆θ V .
(17)
Any process coinciding with the (G, Q) intensity of θ before θ is an eligible
process h in their setup

44 / 36

In a basic immersed setup, E∆θ V vanishes and equality holds in (15): see
the comments before Section 3 in Duffie et al. (1996), page 1379 in
Collin-Dufresne et al. (2004), or following (3.22), (H.3) and Proposition 6.1
in Bielecki and Rutkowski (2001)).
But, in general, E∆θ V is nonnull and intractable.

45 / 36

One specific instance of (13), which corresponds to the approach by
Collin-Dufresne et al. (2004, Theorem 1), consists in using m∗ = m andW ∗ = WR, taking for Q∗ the so-called survival measure1 with (G, Q) density·
process e 0 γ(u,mu ,(0,u))du 1(0,θ] .
This fixes the discrepancy in (15) by singularly changing the probability
measure, while sticking to the original model filtration Gor, more precisely, resorting to the Q∗ augmentation G∗ of G, obtained byadding to G all the Q∗ null sets A such that A ⊆ {θ ≤ T }).

1 The “survival measure” idea and terminology were first introduced and used for various
purposes in Schönbucher (1999, 2004)).
46 / 36

Another instance of (13) consists in usingm∗ = m and dWt∗ = dBt − µ(t, mt , (0, t))dt.
As it follows from Lemma 3.5 and Section 4.4 in Crépey and Song (2017),
this process W ∗ is a (G∗ = F = B, Q∗ = P) Brownian motion, for someprobability measure P, distinct from Q but equivalent to it on FT , on which
P is uniquely determined through (6).

47 / 36

The corresponding formula (14) is none other than our expectation formula(7).
This approach fixes the discrepancy in (15) by reducing the filtration from G
to F = B, while changing the probability measure “as little as possible”,
i.e. equivalently on FT
In a basic immersive setup, an invariance time approach does not change Q atall, whereas Collin-Dufresne et al. (2004)’s measure change is still singular

48 / 36

We emphasize that Collin-Dufresne et al. (2004)’s approach only provides atransfer of conditional expectation formulas (because of the singularity of
their measure change), as opposed to a transfer of semimartingale calculus asa whole under an invariance time approach.
From a more general and theoretical perspective, these different approaches
can be related via the generalized Girsanov formulas of Kunita (1976) andYoeurp (1985) (cf. Song (2013)).

49 / 36

Bielecki, T. and M. Rutkowski (2001).
Credit risk modelling: Intensity based approach.
In E. Jouini, J. Cvitanic, and M. Musiela (Eds.), Handbook in Mathematical
Finance: Option Pricing, Interest Rates and Risk Management, pp.
399–457. Cambridge University Press.
Collin-Dufresne, P., R. Goldstein, and J. Hugonnier (2004).
A general formula for valuing defaultable securities.
Econometrica 72 (5), 1377–1407.
Crépey, S. and S. Song (2017).
Invariance properties in the dynamic Gaussian copula model.
ESAIM: Proceedings and Surveys 56, 22–41.
Preprint on https://math.maths.univ-evry.fr/crepey.
Duffie, D., M. Schroder, and C. Skiadas (1996).
Recursive valuation of defaultable securities and the timing of resolution of
uncertainty.
The Annals of Applied Probability 6 (4), 1075–1090.
Jeulin, T. and M. Yor (1978).
Grossissements de filtrations et semi-martingales: formules explicites.
50 / 36

In Séminaire de Probabilités, Volume XII of Lecture Notes in Mathematics 649,
pp. 78–97. Springer.
Kunita, H. (1976).
Absolute continuity of markov processes.
In Séminaire de Probabilités de Strasbourg X, 1974/75, Lecture Notes Math.
511, pp. 44–77. Springer.
Nikeghbali, A. and M. Yor (2005).
A definition and some characteristic properties of pseudo-stopping times.
Annals of Probability 33, 1804–1824.
Schönbucher, P. (1999).
A Libor market model with default risk.
Working Paper, University of Bonn.
Schönbucher, P. (2004).
A measure of survival.
Risk Magazine 17 (8), 79–85.
Song, S. (2013).
Local solution method for the problem of enlargement of filtration.
arXiv:1302.2862.
51 / 36Song, S. (2014).
Local martingale deflators for asset processes stopped at a default time s t orjust before s t− .
arXiv:1405.4474.
Yoeurp, C. (1985).
Grossissements de filtrations: exemples et applications.
In Théorème de Girsanov généralisé et grossissement d’une filtration, Lecture
Notes Math. 1118, pp. 172–196. Springer.

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